Question

calculating geometric probability
complete the steps to find the probability that a randomly chosen point on the triangle is in the shaded rectangle.
the area of the triangle is cm²?
the area of the rectangle is cm²?
the probability that the point is in the shaded area is (options: 0.32, 0.47, 0.48, 0.68).
diagram: right triangle with base 20 cm, height 15 cm; shaded rectangle with length 8 cm, height 6 cm.

Explanation:

Step1: Calculate area of triangle

The formula for the area of a right - triangle is $A=\frac{1}{2}\times base\times height$. Here, the base of the triangle is $20$ cm and the height is $15$ cm. So, $A_{triangle}=\frac{1}{2}\times20\times15$.
$\frac{1}{2}\times20\times15 = 10\times15=150$ $cm^{2}$.

Step2: Calculate area of rectangle

The formula for the area of a rectangle is $A = length\times width$. Here, the length of the rectangle is $8$ cm and the width is $6$ cm. So, $A_{rectangle}=8\times6$.
$8\times6 = 48$ $cm^{2}$.

Step3: Calculate the probability

The probability $P$ that a randomly chosen point in the triangle is in the shaded rectangle is the ratio of the area of the rectangle to the area of the triangle. So, $P=\frac{A_{rectangle}}{A_{triangle}}=\frac{48}{150}$.
Simplify $\frac{48}{150}=\frac{8}{25}=0.32$.

Answer:

• The area of the triangle is $\boldsymbol{150}$ $cm^{2}$.
• The area of the rectangle is $\boldsymbol{48}$ $cm^{2}$.
• The probability that the point is in the shaded area is $\boldsymbol{0.32}$.